The geodetic boundary value problem in two dimensions and its iterative solutions
Martin van Gelderen
Publications on Geodesy 35,
Delft, 1991. 143 pages.
ISBN-13: 978 90 6132 241 2. ISBN-10: 90 6132 241 3. € 10.50
Abstract
In this thesis, the geodetic boundary value problem (GBVP) for a
completely hypothetical earth is developed. As already shown in (Gerontopoulos,
1978) the complete GBVP for a 2D earth can be set up. It can serve as an
example for the real 3D one, with the advantage of less complex
mathematics and better performable numerical simulations.
In chapter one, the points of departure of this thesis are discussed. It
is underlined that we do not seek for a strict mathematical solution of
the GBVP, as done by Gerontopoulos, but investigate aspects of the 2D
GBVP that have correspondence with the 3D case. This introductory
chapter is concluded by an overview of the history of the problem of the
determination of the figure of the earth.
Chapter two serves as preparation for the formulation and solution of the
GBVP. Some points of the potential theory in the plane are treated and
special attention is paid to the series solution of the potential for a
circular boundary, which is an ordinary Fourier series. Finally expressions
are derived for the integral kernels appearing in the solution of the GBVP.
In chapter three, the linear observation equations are derived for the
classical observations potential, gravity and astronomical latitude, and for
the components of the gravity gradient tensor. From several combinations of
these observables, the potential, and the position are solved with the
observation equations in circular, and constant radius approximation. For
their solution, closed integral expressions are given. The systems of
equations can be either uniquely determined or overdetermined. This yields
solutions for the disturbing potential which are almost identical to the 3D
problem. It is also possible to solve, in this approximation, the GBVP
analytically from discrete measurements. The analytical expression derived
for the inverse normal matrix can be used for error propagation. It is shown
that the integral of astronomical leveling can be derived from the solution
of the GBVP with observations of astronomical latitude. Furthermore,
attention is paid to the zeroth and first degree coefficients, and to the
application of the theory of reliability to the GBVP.
In chapter four, first the effect of the neglect of the topography and the
ellipticity is analysed. It follows an iteration method can be applied in
order to obtain solutions of the GBVP without, or with only little,
approximation. Then, five levels of approximation are defined: three linear
approximations (with or without the topography and/or the ellipticity taken
into account), a quadratic model and the exact, non-linear equations. In the
iteration the analytical solutions of the GBVP'S in circular, constant
radius approximation are used for the solution step. For the backward
substitution the model is applied for which the solution is sought for. The
problems are solved numerically by iteration in chapter five. The iterative
solution of the problem in circular approximation, occasionally referred to
as the simple problem of Molodensky, is also given as a series of integrals.
For the convergence of
the iteration criteria are derived.
In chapter five, the generation of a synthetical world is presented. The
features of the real world, with respect to the topography and the gravity
field, are used to determine its appearance. The observations are computed,
from which the potential and the position are solved by the iteration, for
all five levels of approximation defined. The fixed, scalar and vectorial
problem are considered. It turns out that, in case band limited observations
without noise are used, the ellipticity of the earth,
not taken into account in the solution step of the iteration, is the main
obstacle for convergence. This can be overcome by the use of a potential
series with elliptical coordinates, instead of the polar coordinates usually
applied. The theoretical condition for convergence of the iteration is
tested, and for several circumstances the accuracy of the solution of the
potential and position unknowns is computed. We mention: uniquely determined
and overdetermined problems, band limited observations, block averages and
point values, number of points etc. Finally, the error spectra of the solved
coefficients are compared to the error estimates obtained by error
propagation with the analytical expression for the inverse normal matrix of
the GBVP in circular, constant radius approximation, and a simple noise
model for the observations. If the data noise is the dominant error source,
this error estimation turns out to work very well.
Contents
Abstract v
Acknowledgments vii
1. Introduction 1
2. Potential theory of a two-dimensional mass distribution 8
3. The linear geodetic boundary value problem by least squares 2 0
4. Higher order approximations of the linear problem 59
5. Numerical experiments 81
Conclusions 122
A. Coordinate frames and their transformations 124
B. Elliptical harmonics 132
References 140